The Concept of Derivative and Basic Properties

5.1 The Concept of Derivative and Basic Properties

Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. [“Given an equation which involves the derivatives of one or more functions, find the functions.”] Sir Isaac Newton to G.W. Leibniz, October 24, 1676

The above quotation is the decipherment of an anagram in which Newton points out that differential equations are important because they express the laws of

nature. 1 It is now fully recognized that the origins of the differential calculus go back 1 Newton sent his letter to Henry Oldenburg, secretary of the Royal Society, and through him to G.W. Leibniz in Germany. Oldenburg was later imprisoned in the Tower of London for correspond-

ing with foreigners.

184 5 Differentiability to the works of Sir Isaac Newton (1643–1727) and Gottfried Wilhelm von Leibniz

(1646–1716). However, when calculus was first being developed, there was a con- troversy as to who came up with the idea “first.” It is thought that Newton had discovered several ideas related to calculus earlier than Leibniz. However, Leibniz was the first to publish. Today, both Leibniz and Newton are considered to have discovered calculus independently.

The concept of derivative has been applied in various domains and we mention in what follows only some of them: – velocity and acceleration of a movement (Galileo 1638, Newton 1686); – astronomy, verification of the law of gravitation (Newton, Kepler); – calculation of the angles under which two curves intersect (Descartes); – construction of telescopes (Galileo) and clocks (Huygens 1673); – search for the maxima and minima of a function (Fermat 1638).

Let f be a real function with domain I ⊂ R. If x 0 ∈ I is an interior point of I, then the limit

lim (x) − f (x 0

x −x 0

when it exists, is called the derivative of f at x 0 . In this case we say that f is dif- ferentiable at x 0 and the value of the above limit is called the derivative of f at x 0 and is denoted by f ′ (x 0 ). If I is an open interval and if f is differentiable at every x 0 ∈ I, then we say that f is differentiable on I. We point out that the notation f ′ (x 0 ) is a descendant of notation introduced by Newton, while Leibniz used the notation (d f /dx)(x 0 ). The nth-order derivative of f is denoted by f (n) . We notice that the Newton quotient

f (x) − f (x 0 ) x −x 0

Tangent at x 0

f(x)

f(x 0 )

Chord from x 0 and x

Fig. 5.1 Tangent and chord.

5.1 The Concept of Derivative and Basic Properties 185

Fig. 5.2 Graph of the function f (x) = √ 3 x .

measures the slope of the chord of the graph of f that connects the points (x, f (x)) and (x 0 , f (x 0 )) (see Figure 5.1). The one-sided derivatives of f at x 0 (if they exist) are defined by

Moreover, f is differentiable at x 0 if and only if f ′ (x 0 −) and f ′ (x 0 +) exist, are finite, and f ′ (x 0 −) = f ′ (x 0 +). The point x 0 is said to be an angular point if f ′ (x 0 −) and f ′ (x 0 +) are finite and f ′ (x 0 ′ (x 0 +). We say that x 0 is a cusp point of f if f ′ (x 0 −) = ±∞ and f ′ (x 0 +) = ∓∞ (see Figure 5.2), while we say that f has a vertical tangent at x 0 if f ′ (x 0 −) = f ′ (x 0 +) = ±∞ (see Figure 5.3). The above definition implies that if f is differentiable at x 0 , then f is continuous at x 0 . The converse is not true, and the most usual counterexample in this sense is given by the mapping f : R →R defined by f (x) = |x| (see Figure 5.4). Indeed, this function is continuous at the origin, but is not differentiable at x 0 = 0. Moreover,

Fig. 5.3 Graph of the function f (x) = |x|.

Fig. 5.4 Graph of the function f (x) = |x|.

f ′ (0−) = −1 and f ′ (0+) = +1. A function f : R→R that is everywhere continuous but nowhere differentiable is

f (x) = ∑ 2 −n sin n 2 x for all x ∈ R.

n =1

In fact, the first example of a continuous function that is nowhere differentiable is due to Weierstrass (1872), who shocked the mathematical world by showing that the continuous function

f (x) =

b n cos (a ∑ n x ) (0 < b < 1)

n =1

is nowhere differentiable, provided ab >1+3 π /2. The function

is discontinuous for all x ′ (0) = 1.

A related notion is the following. Let I ⊂ R be an interval. We say that a function

f :I →R has a symmetric derivative in x 0 ∈ IntI if there exists

If there exist f ′ (x 0 +) and f ′ (x 0 −), then there exists the symmetric derivative f ′ s (x 0 ) and, moreover,

The converse is not true, as shown by the following counterexample:

x sin 1

f (x) =

if x =0.

5.1 The Concept of Derivative and Basic Properties 187 We have already seen that Lipschitz maps are continuous. Of course, they do not

need to be differentiable, as show by the absolute value map f (x) =| x | (x ∈ R), which satisfies

| f (x) − f (y) |≤| x − y | for all x , y ∈ R,

but which is not differentiable at the origin. The following question arises naturally: are there differentiable functions that are not Lipschitz? The answer is yes, and a function with such a property is

In 1872, Weierstrass shocked the mathematical world by giving an example of a function that is continuous at every point but whose derivative does not exist any- where . Consider the mapping ϕ :R →R defined by

x − n, ϕ if n is even and n ≤ x < n + 1,

∑ n ϕ (4 x ).

n =1

Then f defined as above is continuous at every real x but differentiable at no real x . The celebrated French mathematician Henri Poincar´e [L’oeuvre math´ematique

de Weierstrass , Acta Math., vol. 22 (1899)] mentioned about this striking example, “A hundred years ago such a function would have been considered an outrage on common sense.”

The derivative of a function is directly relevant to finding its maxima and minima. We recall that if f is a real function defined on an interval I, then a point x 0 ∈ I is called a local minimum for f if there exists δ > 0 such that (x 0 − δ ,x 0 + δ ) ⊂ I and

f (x) ≥ f (x 0 ) for all x ∈ (x 0 − δ ,x 0 + δ ). If x 0 is a local minimum of the function − f , we say that x 0 is a local maximum of f .

The following Fermat test is an important tool for proving inequalities, and since it holds even in higher dimensions, it also applies in critical-point theory, in order to prove the existence of weak solutions to partial differential equations. This theorem was discovered by the French mathematician Pierre de Fermat (1601–1665) and

expresses a geometrically intuitive fact: if x 0 is an interior extremal point for f in

I and if the graph of f has a tangent line at (x 0 , f (x 0 )), then such a line must be parallel to the x-axis. Fermat’s Theorem. Let f be real-valued function defined on an interval I . If f has a local extremum at an interior point x 0 of I and if f is differentiable at x 0 , then

f ′ (x 0 )=0 .

A more precise answer can be formulated in terms of the second derivative. More exactly, if f : I →R has a local maximum (resp., local minimum) at some interior

point x 0 of I, then f ′′ (x 0 ) ≤ 0 (resp., f ′′ (x 0 ) ≥ 0).

188 5 Differentiability Mean value theorems play a central role in analysis. The simplest form of the

mean value theorem is stated in the next basic result, which is due to Michel Rolle (1652–1719).

Rolle’s Theorem. Let f be a continuous real-valued function defined on the closed interval [a, b] that is differentiable on (a, b) . If f (a) = f (b) , then there

exists a point c ∈ (a,b) such that f ′ (c) = 0 .

Geometrically, Rolle’s theorem states that if f (a) = f (b) then there is a point in the interval (a, b) at which the tangent line to the graph of f is parallel to the x -axis (see Figure 5.5). Another geometrical implication of Rolle’s theorem is the following.

Polar Form of Rolle’s Theorem. Assume that f is a continuous real-valued function, nowhere vanishing in [ θ 1 , θ 2 ] , differentiable in ( θ 1 , θ 2 ) , and such that

f ( θ 1 )=f( θ 2 ) . Then there exists θ 0 ∈( θ 1 , θ 2 ) such that the tangent line to the graph r =f( θ ) at θ = θ 0 is perpendicular to the radius vector at that point. Taking into account the geometric interpretation of Rolle’s theorem, we expect that is would be possible to relate the slope of the chord connecting (a, f (a)) and (b, f (b)) with the value of the derivative at some interior point. In fact, this is the main content of a mean value theorem. The following is one of the basic results in elementary mathematical analysis and is known as the mean value theorem for differential calculus [Joseph-Louis Lagrange (1736–1813)]. Geometrically, this the- orem states that there exists a suitable point (c, f (c)) on the graph of f : [a, b]→R such that the tangent is parallel to the straight line through the points (a, f (a)) and (b, f (b)) (see Figure 5.6).

Lagrange’s Mean Value Theorem. Let f be a continuous real-valued function defined on the closed interval [a, b] that is differentiable on (a, b) . Then there exists

a point c ∈ (a,b) such that

Fig. 5.5 Geometric illustration of Rolle’s theorem.

5.1 The Concept of Derivative and Basic Properties 189

Fig. 5.6 Geometric illustration of Lagrange’s theorem.

A simple statement that follows directly from the above result is that any differ- entiable function with bounded derivative is Lipschitz (compare with the comments at the beginning of this chapter!).

A useful consequence of the Lagrange mean value theorem shows that computing the sign of the first derivative of a differentiable function suffices to localize maxima and minima of the function . More precisely, if f is differentiable on an open interval

I , then f is increasing on I if and only if f ′ (x) ≥ 0 for all x ∈ I. This property is also called the increasing function theorem. Furthermore, if f ′ (x) > 0 for all x ∈ I, then f is strictly increasing in I. A direct consequence of this property is that if

f ′ (x) ≤ g ′ (x) on [a, b], then f (x) − f (a) ≤ g(x) − g(a) for all x ∈ [a,b]. This result is sometimes refereed as the racetrack principle: if one car goes faster than another, then it travels farther during any time interval.

The converse of the property

f ′ > 0 in (a, b) =⇒ f strictly increasing in (a,b)

is not true. Indeed, the function x (2 − cos(lnx) − sin(lnx)) if x ∈ (0,1],

f (x) =

0 if x = 0, is (strictly!) increasing on [0, 1], but there are infinitely many points ξ ∈ (0,1) such

that f ′ ( ξ ) = 0 (see Figure 5.7). The definition of differentiability implies that if f : (a, b)→R is differentiable at x 0 ∈ (a,b) and f ′ (x 0 ) > 0, then there exists δ > 0 such that

f (x) > f (x 0 ) for all x ∈ (x 0 ,x 0 + δ ),

f (x) < f (x 0 ) for all x ∈ (x 0 − δ ,x 0 ).

Fig. 5.7 A strictly increasing function with derivative vanishing infinitely many times.

The above statement does not imply that a function satisfying f ′ (x 0 ) > 0 is incr- easing in a neighborhood of x 0 . As a counterexample, consider the function (see Figure 5.8)

This function is differentiable everywhere and satisfies f ′ (0) = 1. However, for x

(x) = 1 + 2x sin 2 − cos

oscillates strongly near the origin (see Figure 5.9). Hence, even though the graph of

f is contained between the parabolas y 1 =x−x 2 and y 2 =x+x 2 , there are points with negative derivative arbitrarily close to the origin. If f ′ (x) > 0 for all x ∈ (a,b),

0 0.1 0.2 0.3 0.4 Fig. 5.8 Graph of the function f (x) = x + x 2 sin (1/x 2

5.1 The Concept of Derivative and Basic Properties 191

Fig. 5.9 Graph of the function R

x 2 − 2 x cos 1 x 2 .

then f is increasing. Thus, this counterexample is possible only because f is not continuously differentiable.

A much more elaborated notion of monotony is that of a completely monotonic function. This is a real-valued function defined on an interval I of the real line that satisfies

(−1) n f (n) (x) > 0 for all x ∈ I and n ∈ N ∗ . Such functions do exist. An √ example is given by the mapping f : (1, ∞)→R defined by f (x) = x 2 − 1. An

induction argument shows that for all x > 1, f (n) (x) > 0 for odd n and f (n) (x) < 0 for even n > 0.

Other immediate consequences of the mean value theorem establish that if f is a real-valued function that is continuous on [a, b] and differentiable on (a, b), then the following hold:

(i) if f ′

for all x ∈ (a,b) then f is one-to-one on [a,b];

(ii) if f ′ (x) = 0 for all x ∈ (a,b) then f is constant on [a,b]. The following example (devil’s staircase) shows that the last result is not as

trivial as it might appear. If x ∈ [0,1] has a representation in base 3 as, e.g., x = 0.2022002101220 . . ., then f (x) is obtained in base 2 by converting all 2’s preceding the first 1 to a 1 and deleting subsequent digits. In our case, f (x) =

0 .101100011. In particular, f (x) = 1/2 on [1/3, 2/3], f (x) = 1/4 on [1/9, 2/9],

f (x) = 3/4 on [7/9, 8/9], etc. This function f is continuous and nondecreasing. Moreover, f is differentiable with derivative f ′ (x) = 0 on a set of measure

hence almost everywhere. Nevertheless, f

A useful consequence of the Lagrange mean value theorem is the following. Corollary. Let f : [a, b)→R

be a differentiable function that is continuous at x =a and such that lim x ցa f ′ (x) exists and is finite. Then f is differentiable at x =a and f ′ (a) = lim x ցa f ′ (x) .

192 5 Differentiability

A generalization of the Lagrange mean value theorem is stated in the next result, which is due to the celebrated French mathematician Augustin-Louis Cauchy (1789–1857).

Cauchy’s Mean Value Theorem. Let f and g be continuous real-valued func- tions defined on [a, b] that are differentiable on (a, b) and such that g . Then there is a point c ∈ (a,b) such that

f (b) − f (a)

f ′ (c) = ′

g (b) − g(a)

g (c)

A geometric interpretation of Cauchy’s mean value theorem is that if p (x) = ( f (x), g(x)) is a path in the plane, then under certain hypotheses, there must

be an instant at which the velocity vector is parallel to the vector joining the end- points of the path. We point out that the mean value theorem is no longer valid if f takes values in R N , with N ≥ 3. The mean value theorem is still valid in R 2 provided one states it correctly. Specifically, say that two vectors v and w are parallel if there is some non- trivial linear relation av + bw = 0, that is, if the span of v and w has dimension less than 2. Note that this is not an equivalence relation, since the zero vector is parallel

to anything. Then for any C 1 function f : [a, b]→R 2 there is some c ∈ (a,b) such that f ′ (c) and f (b) − f (a) are parallel. The result fails in R 3 and, more generally, in any R N , where N ≥ 3. Indeed, consider the function f : [0,2 π ]→R 3 defined by

f (x) = (cos x, sin x, x). Then f (2 π ) − f (0) = (0,0,2 π ) but f ′ (x) = (−sinx,cosx,1), for all x ∈ [0,2 π ]. In fact, the authentic sense of the mean value theorem is that of mean value inequality. For instance, if f is a real-valued function such that m ≤f ′ (x) ≤ M on the interval [a, b], then

m (x − a) ≤ f (x) − f (a) ≤ M(x − a),

for all x in [a, b]. It is interesting to point out that before Lagrange and Cauchy, Amp`ere [34] saw the importance of the above mean value inequality and even used it as the defining property of the derivative.

The following result, discovered by Gaston Darboux (1842–1917), asserts that the derivative of a real-valued function defined on an interval has the intermediate value property.

Darboux’s Theorem. Let I be an open interval and let f :I →R

be a differen-

tiable function. Then f ′ has the intermediate value property.

If f ′ is continuous, then Darboux’s theorem follows directly from the intermedi- ate value property of continuous functions. But a derivative need not be continuous. Indeed, the function

f (x) = · sin x

if x = 0, is differentiable at x = 0 but lim x → 0 f ′ (x) does not exist (see Figure 5.10).

5.1 The Concept of Derivative and Basic Properties 193

Fig. 5.10 A differentiable function whose derivative is not continuous at x = 0.

The standard proof of Darboux’s theorem is based on the fact that a continuous function on a compact interval has a maximum. This proof relies on considering the auxiliary function g (t) = f (t) − y 0 t , where a , b ∈ I are fixed and y 0 lies strictly between f ′ (a) and f ′ (b). Without loss of generality assume a < b and f ′ (a) > y 0 >

f ′ (b). Then g ′ (a) > 0 > g ′ (b), so a and b are not local maxima of g. Since g is continuous, it must therefore attain its maximum at an interior point x 0 of [a, b]. So, by Fermat’s theorem, g ′ (x 0 ) = 0, and this concludes the proof. ⊓ ⊔

The following new proof of Darboux’s theorem is due to Olsen [86] and is based on an interesting approach that uses the Lagrange mean value theorem and the inte- rmediate value theorem for continuous functions. We present this elegant proof in

what follows. Assume that y 0 lies strictly between f ′ (a) and f ′ (b). Define the con- tinuous functions f a ,f b :I →R by

f ′ (a)

for t = a,

f a (t) = f (t)− f (a)

f b (t) = f (t)− f (b)

t −b

for t

Then f a (a) = f ′ (a), f a (b) = f b (a), and f b (b) = f ′ (b). Thus, either y 0 lies between

f a (a) and f a (b), or y 0 lies between f b (a) and f b (b). In the first case, by the continu- ity of f a , there exists s ∈ (a,b] such that

The Lagrange mean value theorem implies that there exists x 0 ∈ (a,s) such that

(s) − f (a)

=f ′ (x ).

−a 0

194 5 Differentiability In the second case, assuming that y 0 lies between f b (a) and f b (b), a similar arg-

ument based on the continuity of f b shows that there exist s ∈ [a,b) and x 0 ∈ (s,b) such that y 0 = [ f (s) − f (b)] /(s − b) = f ′ (x 0 ). This completes the proof. ⊓ ⊔ The significance of Darboux’s theorem is deep. Indeed, since f ′ always satisfies the intermediate value property (even when it is not continuous), its discontinuities are all of the second kind. Geometrically, Darboux’s theorem means that although derivatives need not be continuous, they cannot suffer from jump discontinuities.

Another important result is Taylor’s formula, which allows us to find approxi- mate values of elementary functions. It was discovered by the British mathematician Brook Taylor (1685–1731). We first recall that if f : (a, b)→R is n times differen-

tiable at some point x 0 ∈ (a,b), then the Taylor polynomial of f about x 0 of degree n is the polynomial P n defined by

Often the Taylor polynomial with center x 0 = 0 is called the Maclaurin polynomial [A Treatise of Fluxions, Colin MacLaurin (1698–1746)]. Taylor’s Formula with Lagrange’s Reminder. Let f be an arbitrary function in

C n +1 (a, b) and x 0 ∈ (a,b) . Then for all x ∈ (a,b) , there exists a point ξ in the open

interval with endpoints x 0 and x such that

f (n+1) ( ξ )

f (x) − P n (x) = (x − x 0 ) n +1 .

(n + 1)!

Related to this result, in 1797, Lagrange wrote a treatise that allowed him (as he thought) to assert that any C ∞ function f : (a, b)→R is equal to its Taylor series about x 0 ∈ (a,b), that is, it can be expressed as

∞ f (n) (x

f 0 (x) = ) ∑ (x − x 0 ) n .

n =0

Lagrange’s dream lasted some 25 years, until in 1823 Cauchy considered the func- tion f : R →R defined by

e −1/x 2 if x

This function is continuous everywhere, but is extremely flat at the origin (see Figure 5.11), in the sense that f (n) (0) = 0 for all n [Hint: f (n) (x) = p n −1

−1/x 2 , for all x

n denotes a polynomial with integer coefficients]. Thus, the Maclaurin series for the function f (x) is 0 + 0 + ··· and obviously con-

verges for all x. This shows that relation (5.1) is wrong for x 0

A function f satisfying relation (5.1) for all x in some neighborhood of x 0 is called analytic at x 0 . This shows that the function f defined by (5.2) is not analytic at the

5.1 The Concept of Derivative and Basic Properties 195

Fig. 5.11 Graph of the function defined by relation (5.2).

origin, even if its associated Taylor series converges. A related remarkable result of Emile Borel (1871–1956) states that if (a n ) n ≥0 is any real sequence, then there are infinitely many differentiable functions f such that f (n) (0) = a n for all integer n ≥ 0.

An important role in asymptotic analysis is played by the following notation, which is due to Landau:

(i) f (x) = o(g(x)) as x→x 0 if f (x)/g(x)→0 as x→x 0 ;

(ii) f (x) = O(g(x)) as x→x 0 if f (x)/g(x) is bounded in a neighborhood of x 0 ;

(iii) f ∼ g as x→x 0 if f (x)/g(x)→1 as x→x 0 .

Using the above notation, we give in what follows the Taylor expansions of some elementary functions:

=1+x+ +

+ o(x n ) as x→0,

+ o(x n (1 + x) = x − +1 − ··· + (−1) ) as x→0,

2 3 n +1 x 3 x 5 n x 2n +1

sin x + + o(x 2n =x− +2 − ··· + (−1) ) as x→0, 3! 5!

2! ) as x→0,

+ o(x

(2n)!

x 3 2x 5 17x 7 62x 9 1382x 11 21844x 13 tan x =x+ +

6081075 ) as x→0,

1 x 3 1 ·3 x 5 (2n − 1)!! x 2n +1

arcsin x =x+ + + o(x 2n + ··· + +2 ) as x→0,

2 3 2 ·4 5 (2n)!! 2n +1

1 x 3 1 2n +1 arccosx

·3 x 5 (2n − 1)!! x

= −x− −

) as x→0, +1

+ o(x 2n +2

2 2 3 2 ·4 5 (2n)!! 2n

x 3 x 5 x 2n +1

arctan x =x− n +

) as x→0,

2n +1

196 5 Differentiability

x 3 x 5 x 2n +1

sinh x =x+ + + o(x 2n + ··· + +2 ) as x→0, 3! 5!

(2n + 1)!

x 2 x 4 x 2n

cosh x =1+ +

2! 4! + ··· + ) as x→0,

+ o(x 2n +1

(2n)!

x 3 2x 5 17x 7 62x 9 1382x 11 21844x 13 tanh x

=x− 14 + − +

6081075 ) as x→0,

=1+x+x 2 +x 3 n + o(x n

) as x→0,

−x

(1 + x) α =1+C 1 x +C 2 x 2 n x α n α + ··· +C α + o(x n ) as x→0 ( α ∈ R),

where the binomial coefficients are given by

α ( α − 1)( α − 2)···( α − n + 1) α : =

The following theorem (whose “discrete” variant is the Stolz–Ces`aro lemma for sequences) provides us a rule to compute limits of the indeterminate form 0 /0 or ∞/∞. This result is due to Guillaume Franc¸ois Antoine de l’Hˆopital, Marquis de Sainte-Mesme et du Montellier Compte d’Autremonts, Seigneur d’Ouques et autre lieux (1661–1704), who published (anonymously) in 1691 the world’s first textbook on calculus, based on Johann Bernoulli’s notes.

L’Hˆopital’s Rule. Let I be a nonempty open interval in R , let x 0 ∈ R∪{−∞,+∞}

be two differentiable functions, with g strictly monotone and g ′

be one of its endpoints, and let f , g :I →R

for all x ∈I . Assume that either of the next two conditions is satisfied:

(i) f (x) , g (x)→0 as x →x 0 ; (ii) g (x)→ ± ∞ as x →x 0 (no assumptions on f ).

Then the following implication holds:

f (x) lim

f ′ (x)

exists and equals x → x 0 g ℓ. (x) Some important comments related to l’Hˆopital’s rule are stated below.

exists and equals x → x 0 g ′ (x)

ℓ ∈ R ∪ {−∞,+∞} ⇒ lim

(i) We can not assert that if lim x → x 0 f (x)/g(x) exists, then lim x → x 0 f ′ (x)/g ′ (x) exists, too. Indeed, consider the C ∞ –functions

1 −1/x 2 −1/x x 2 sin

x 4 e f e (x) = g (x) =

0 , if x =0. Then lim x → 0 f (x)/g(x) = 0 but

if x = 0,

f ′ (x)

g ′ =− cos x x 4 + o(1) as x→0,

(x)

which implies that lim x → 0 f ′ (x)/g ′ (x) = 0 does not exist.

5.1 The Concept of Derivative and Basic Properties 197 (ii) We are not entitled to draw any conclusion about lim x → x 0 f (x)/g(x) if

lim x → x 0 f ′ (x)/g ′ (x) does not exist. Strictly speaking, if g ′ has zeros in every neighborhood of x 0 , then f ′ /g ′ is not defined around x 0 and we could assert that lim x → x 0 f ′ (x)/g ′ (x) does not exist. Stolz [108] discovered a strik- ing phenomenon related to the applicability of l’Hˆopital’s rule. More precisely,

he showed that it is possible that f ′ and g ′ contain a common factor, that is,

f ′ (x) = A(x) f 1 (x) and g ′ (x) = A(x)g 1 (x), where A(x) does not approach a limit, but lim x → x 0 f 1 (x)/g 1 (x) exists. This means that under these circumstances, lim x → x 0 f ′ (x)/g ′ (x) may exist but lim x → x 0 f (x)/g(x) not exist. For instance, let us consider the example given by Stolz:

x → ∞ g ′ (x) x → ∞ cos xe sin x (x + 2 cosx + sin x cos x)

2 cos x

= lim

= 0. x → ∞ e sin x (x + 2 cosx + sin x cos x)

However, lim x ∞ f (x)/g(x) does not exist. This is due to the fact that g → ′ has zeros in every neighborhood of ∞, and consequently, we are not entitled to apply l’Hˆopital’s

rule. (ii) Let f be differentiable on an interval containing x = a. Then

and (since f is continuous at x = a) both the numerator and the denominator of the difference quotient approach zero. L’Hˆopital’s rule then gives

a result that apparently shows that f ′ is continuous at x = a. However, taking the function

we observe that not every differentiable function has a continuous derivative. The student should understand that because of the hypothesis of l’Hˆopital’s rule, relation

(5.3) is true in the sense that “if lim x → a f ′ (x) exists, then lim x → a f ′ (x) = f ′ (a).” We conclude this section with the following useful result.

Differentiation Inverse Functions Theorem. Suppose f is a bijective differen-

for all x ∈ [a,b] . Then f −1 exists and is differentiable on the range of f and, moreover, (f −1 ) ′ [ f (x)] = 1/ f ′ (x) for all x ∈ [a,b] .

tiable function on the interval [a, b] such that f ′

198 5 Differentiability